Let $f$ be analytic in $\Omega_\rho$ and continuous on $\overline{\Omega}_\rho$, with $\rho>1$, and let $I_Nf$ be its Chebyshev-Lobatto interpolant. Then for each fixed derivative order $m\ge0$, there are constants $C_m>0$ and $c_m>0$, depending on $f$, $m$, and $\rho$, such that \begin{align*} \|f^{(m)}-(I_Nf)^{(m)}\|_\infty\le C_m N^{c_m}\rho^{-N}. \end{align*}